Logarithmic Correction to BPS Black Hole Entropy from Supersymmetric Index at Finite Temperature

It has been argued by Iliesiu, Kologlu and Turiaci in arXiv:2107.09062 that one can compute the supersymmetric index of black holes using black hole geometry carrying finite temperature but a specific complex angular velocity. We follow their prescription to compute the logarithmic correction to the entropy of BPS states in four dimensions, defined as the log of the index of supersymmetric black holes, and find perfect agreement with the previous results for the same quantity computed using the near horizon AdS 2 × S 2 geometry of zero temperature black holes. Besides giving an independent computation of supersymmetric black hole entropy, this analysis also provides a test of the procedure used previously for computing logarithmic corrections to Schwarzschild and other non-extremal black hole entropy.


Introduction
Logarithmic corrections to extremal, supersymmetric black hole entropy has provided a stringent consistency test of string theory.In particular, logarithmic corrections to supersymmetric black hole entropy in N = 4 and N = 8 supersymmetric string theories in four dimensions, computed from gravitational path integral over the near horizon AdS 2 × S 2 geometry [1,2], agrees with the result of exact counting in the corresponding string theories [3][4][5][6][7][8][9][10][11]. 1 Similar agreement also holds for supersymmetric black holes in higher dimensional string theories [12].
A summary of some of the results can be found in table 1 of [13].This analysis has also been extended to exponentially suppressed contribution to the indices and for twisted indices [14,15] and to black holes in asymptotically anti de Sitter spaces [16][17][18].
In a relatively recent paper, Iliesiu, Kologlu and Turiaci have suggested that we can compute the index of supersymmetric black holes in flat space-time by working in a finite temperature black hole geometry [19].(Some earlier work along the same line on black holes in AdS spacetime can be found in [20] and on near extremal black holes can be found in [21].)Typically at finite temperature the Euclidean path integral around a black hole geometry is expected to compute the partition function in the grand canonical ensemble.In particular the angular velocity ⃗ Ω of the black hole provides the chemical potential dual to the angular momentum ⃗ J carried by the black hole, and the asymptotic values of the time component µ i of the i-th gauge field provide the chemical potential dual to the electric gauge charge.Therefore the gravitational path integral computes where the trace is taken over all states at fixed magnetic charges ⃗ P .The components of ⃗ P are fixed by the fluxes of the magnetic fields at infinity.In contrast, a supersymmetric black hole at zero temperature develops a near horizon AdS 2 × S 2 geometry and the Euclidean path integral over this geometry gives the degeneracy / index of the black hole in the microcanonical ensemble [22] where the electric and the magnetic charges, as well as the angular momentum, are fixed.
The main idea behind [19] is that if we work at fixed β, µ i and set β ⃗ Ω = (0, 0, −2πi) in (1.1), then the part inside the trace involving the angular momentum becomes e 2πiJ 3 = (−1) F since J 3 is integer (integer + half) for bosons (fermions).Hence the path integral will now measure the supersymmetric index. 2 Since supersymmetric index receives contribution only from BPS states that carry lowest possible energy for a given charge, the index computation behaves as if we are at zero temperature even if we keep β finite.Indeed, it was shown in [19] that requiring the solution to have β ⃗ Ω = (0, 0, −2πi) sets the black hole mass M to be its BPS bound.
Therefore even by working with a finite temperature black hole we can study the contribution to the supersymmetric index from BPS black holes.However we still have a sum over electric charges in (1.1).In order to recover the index for a given electric charge from (1.1) we need 2 As we shall see, for the special case when the magnitude of the vector iβ ⃗ Ω is 2π and hence e −β ⃗ Ω. ⃗ J is independent of the direction of ⃗ Ω, there is a family of supersymmetric saddles related by rotation of the vector ⃗ Ω, and we have to integrate over this family to get the correct result for the entropy.
to perform a Laplace transform.In the large charge limit this can be performed using saddle point technique.It was shown in [13] that in four space-time dimensions the Laplace transform does not generate any logarithmic correction to the log of the index.Therefore for computing logarithmic correction to the index we can continue to use the Euclidean gravitational path integral describing (1.1), and fix µ i in terms of Q i 's using the classical solution describing the black hole.
There is, however, an additional subtlety that we need to address.The Witten index of a supersymmetric black hole defined in (1.1) is known to vanish since the black hole always breaks some of the supersymmetries and quantization of the associated goldstino modes produces Bose-Fermi degenerate pair of states.In particular if there are 2n broken supersymmetries then there are 2n such goldstino fermion zero modes.The remedy is to insert n factors of 2J 3 carried by the state into the trace so that the cancellation is prevented for states that break only 2n supersymmetries, but continue to hold for states that break more than 2n supersymmetries [23,24].This small modification of the prescription of [19] will be needed for our analysis.Our goal will be to compute the logarithmic correction to the supersymmetric black hole entropy following the prescription given above and compare the result with the zero temperature computation based on the near horizon AdS 2 × S 2 geometry [1,2,[25][26][27][28].We find perfect agreement between the two computations.In the following we shall refer to the full black hole geometry at finite temperature as the finite temperature geometry and the near horizon AdS 2 × S 2 geometry of a zero temperature black hole as the zero temperature geometry.
The rest of the paper is organized as follows.In section 2 we describe the Kerr-Newman solution in the scaled variable that makes it easier to extract the dependence of various quantities on the overall scale of the charge carried by the black hole.We also review how to normalize the integration measure over various fields in the gravitational path integral.In section 3 we discuss a pair of rotational zero modes of four dimensional Euclidean black holes that arise specifically for the special choice of β ⃗ Ω = (0, 0, −2πi).These are related to the freedom of rotating ⃗ Ω as mentioned in footnote 2. In section 4 we describe the computation of logarithmic corrections to the entropy of supersymmetric black holes in N ≥ 2 string theories.We conclude in section 5 with some comments on our result.Throughout this paper we shall work in the ℏ = c = G = 1 units.

Review
In this section we shall review some of the background material needed for our analysis.

The Kerr-Newman solution in scaled variables
We begin by writing down the Kerr-Newman solution: In this geometry the outer and the inner horizons are located at The inverse temperature β, the angular velocity Ω 3 at the horizon and the chemical potential µ are given by: (2.3) We shall see later that the constant additive term in the expression for A µ dx µ in (2.1) is needed to get a non-singular gauge field configuration at the horizon of the Euclidean black hole.For now we note that the asymptotic value of A t determines the chemical potential µ.
We shall consider an Euclidean version of this black hole and work with scaled variables and coordinates defined as follows: where a is an arbitrary parameter.In this case the solution takes the form: and we have (2.7) The horizon corresponds to the surface ρ = ρ + .To check the regularity at the horizon, we write the metric as

8)
At ρ = ρ + the coefficient of dρ 2 term blows up while the coefficient of the dτ − b sin 2 θdϕ 2 term vanishes.We now define and express the metric and the gauge fields near the horizon as We have dropped the terms of order ρ2 dτ d φ and ρ2 d φ2 since we shall argue shortly that they represent non-singular terms in the metric near the horizon.We now see that the ρ-τ space describes a smooth disk if we treat τ ≡ τ m 2 − q 2 + b 2 /(ρ 2 + − b 2 ) as an angular coordinate with period 2π, since in the coordinate system x = ρ cos τ , ỹ = ρ sin τ the metric is proportional to dx 2 + dỹ 2 .Also in this coordinate system the ρ2 d φ2 and ρ2 dρd φ terms vanish at x = ỹ = 0.This shows that in order to get a non-singular metric near the horizon, we need to make the identification: This can be expressed as, (τ, ϕ) The constant additive term in A µ dx µ ensures that the integral of A µ dx µ along the contractible cycle parametrized by τ vanishes at the horizon.
As reviewed in the introduction, the saddle point that contributes to the index has β Ω 3 = −2πi.From (2.6) and (2.7) we see that this can be achieved by choosing, This gives

.15)
We can take the large charge limit by taking a to be large at fixed m(= q) and b.With the choice of variables made in (2.4), the metric given in (2.10) has an overall factor of a 2 and the gauge field given in (2.11) has an overall factor of a.This makes it easy to extract the a dependence of various quantities.In this limit the curvature invariants at the horizon remain small so that the contribution from the higher derivative corrections to the action are suppressed.As a result, the leading contribution to the entropy is given by the Bekenstein-Hawking formula and the logarithmic corrections are the dominant corrections to this formula.
We also see that since b and q are free parameters, we can work at an arbitrary rescaled temperature γ −1 and chemical potential µ by adjusting b and q.The physical temperature β −1 scale as a −1 in the large a limit.

Integration measure over the zero modes
In this section we shall review the a dependence of the integration measure over the various zero modes (and non-zero modes) following [1,2,12,13].
First consider the case of a vector field A µ .Let us denote by f (n) µ the basis functions for A µ , normalized so that f (n) µ does not have any explicit a dependence.Then we can expand A µ µ .The integration measure DA over the modes c n are fixed so that3 DA e − d4 x √ g g µν AµAν = 1 . (2.16) Using the fact that g µν carries an overall factor of a 2 and f µ is a independent, we see that the terms in the exponent scale as a 2 c m c n .Therefore (2.16) requires that the integration measure over the modes c n must be chosen as (2.17) Next consider the case of metric deformation h µν .Let us denote by f µν the basis functions for h µν , normalized so that f (n) µν does not have any explicit a dependence.Then we can expand µν .The integration measure Dh over the modes h n are fixed so that Dh e − d 4 x √ g g µν g ρσ hµρhνσ = 1 . (2.18) Using the fact that g µν carries an overall factor of a 2 and f µν is a independent, we see that the terms in the exponent scale as h m h n .Therefore (2.18) requires that the integration measure over the modes h n must be chosen as Finally, let us consider the case of gravitino modes.This analysis is similar to that for the gauge field.If g is the a independent basis for expanding the gravitino field ψ µ and if we expand ψ µ as n ψ n g µ , then the integration measure Dψ should satisfy Since the terms in the exponent scale as a 2 ψ † n ψ m , we get (2.21)

Rotational zero modes
Since the Kerr-Newman solution described in section 2.1 carries angular momentum along the z-axis, we can generate zero modes via infinitesimal rotations about the x and y axes.In this section we shall analyze the zero mode generated by rotation about the x-axis.A similar analysis can be carried out for the zero mode associated with rotation about the y-axis.
In the asymptotic region and Cartesian coordinates, the rotation by α x along the x-axis takes the form In polar coordinates this translates to From this it is clear that the range of α x is 0 to 2π, independent of the overall scale a 2 that appears in the expression for the metric.If we replace α x by an arbitrary smooth function of the radial coordinate ρ (and possibly other coordinates) such that it approaches α x as ρ → ∞, then it will generate the same physical configuration as that for constant α x since the two differ by a general coordinate transformation localized in a finite region of space-time.While applying this transformation to the Kerr-Newmann solution, we shall choose the transformation parameter such that it has the form: and smoothly interpolates between 0 and α x in the range ρ 1 < ρ < ρ 0 .Different choices of ρ 0 and ρ 1 are related by local general coordinate transformations and hence describe gauge equivalent solutions.We apply this transformation on the Kerr-Newman solution to generate zero modes.In this case since the deformations vanish at the horizon, we do not need to worry about the regularity of the modes at the horizon.On the other hand, since these transformations change the physical angular momentum of the solution as long as α x ̸ = 0, and since the angular momentum can be expressed as a surface integral at infinity, there will be no general coordinate transformation localized in a finite region of space-time that can reduce the deformed configuration to the original configuration.This ensures that these deformations are genuine deformations and not just gauge artifacts.In the following, the formula for the deformations should be understood as being valid for ρ > ρ 0 .
Therefore this deformation describes genuine zero mode.A similar analysis can be used to show that rotation about the y-axis with parameter α y also generates a normalizable zero mode.
Finally, note that a 2 α x is an overall factor in (3.8).Therefore when we change variables from integration over the modes h n of h µν to the parameters α x , α y , we pick up a jacobian factor of a 2 for each of these modes.The same conclusion is reached if we analyze the modes c n of the gauge fields, since according to (2.17) the integration measure over the modes c n is adc n and we have an overall factor of a in (3.9).The actual zero mode is proportional to a linear combination of the modes h n and a c n as given by (3.8), (3.9).

Logarithmic corrections to the entropy
Our goal in this section will be to compute the logarithmic correction to the entropy of supersymmetric black holes in theories with N ≥ 2 supersymmetry.As was argued in [25], the logarithmic correction to the black hole entropy is not sensitive to the details of the prepotential of the theory and so we can compute this by taking a simple prepotential proportional to (X 0 ) 2 − n V i=1 (X i ) 2 .Furthermore we can assume, without loss of generality, that the black hole carries charge of only the 0-th U(1) gauge field that we identify as the graviphoton of the N = 2 supersymmetric theory.In this case we can construct a black hole solution for which the vector multiplet scalars X i /X 0 remain zero and the solution takes the form of a Kerr-Newman solution.Furthermore, using the electric magnetic duality transformation, we can work with a black hole carrying only electric charge and no magnetic charge.If Q denotes this electric charge then BPS saturation gives M = Q.
For zero temperature, zero angular momentum geometry, the logarithmic correction to the entropy of such black holes was computed in [25,27] by evaluating the integrals over the fluctuations of all the massless fields in the near horizon AdS 2 × S 2 background geometry.Our goal will be to compute this using the finite temperature geometry and then compare the result with that obtained using the zero temperature geometry.For finite temperature, the relevant geometry is the full Kerr-Newman geometry with the angular velocity of the black hole set to (0, 0, −2πi).As discussed in section 2.1, the inverse temperature β is arbitrary.We shall take β ∼ M .In this case we can work in a coordinate system such that under an overall scaling of Q by a, the metric has an overall scale factor a 2 and the area A of the event horizon scales as a 2 .Explicit form of this metric has been given in (2.5).We shall be interested in computing corrections to the black hole entropy of order ln a 2 ∼ ln A, There is one point that needs to be kept in mind in carrying out this comparison.The computation in the zero temperature near horizon geometry gives the result in the microcanonical ensemble while the computation in the finite temperature full Kerr-Newman geometry gives the result in the grand canonical ensemble.Therefore a priori we need to make a Laplace transform before we can compare the two results, since the Laplace transform can generate additional logarithmic terms.However, one can see as follows that in D = 4 we do not need any correction from the change of ensemble.Since the entropy scales as a 2 under the scaling of charges by a in D = 4, the second derivative of the entropy with respect to the charges does not scale with a. Therefore no new logarithmic corrections are generated as we go between the fixed charge and fixed chemical potential system.This can also be seen from eq.(3.12) of [13].
From the finite temperature perspective, since we want to compute the index, we must sum over all angular momentum states and hence as far as the angular momentum is concerned, we do need to work in the grand canonical ensemble.In contrast in the zero temperature near horizon geometry the angular momentum is fixed to zero by requirement of supersymmetry.
The fermion zero modes associated with broken supersymmetry, that are needed to form the supermultiplet carrying different spins, live outside the horizon and produce at most a finite multiplicative factor in the expression for the index.Hence up to this factor, the entire contribution to the index can be expressed in terms of the contribution from zero angular momentum states and this should agree with the result for the index computed using the finite temperature geometry (up to factors of order unity coming from the change of ensemble).

Non-zero mode contributions
The logarithmic terms come from two sources: from path integral over the non-zero modes and path integral over the zero modes.Since the kinetic operator for the bosons has eigenvalues of order a −2 and the kinetic operator for the fermions has eigenvalues of order a −1 , the integration over the zero modes will produce a net contribution of order N B − 1 2 N F ln a = 1 2 N B − 1 2 N F ln A to the entropy, where N B and N F are the total number of bosonic and fermionic non-zero modes respectively.This can be expressed as [29][30][31][32][33][34][35]: where a 4 (x) is the fourth Seeley -DeWitt coefficient [35][36][37] of the massless fields in the theory. 4For minimally coupled charge neutral fields the heat kernel is determined in terms of background Riemann tensor.An example of this is the logarithmic contribution to the entropy due to a minimally coupled scalar analyzed in [38].However, for fields in supergravity that couple to both the background metric and gauge fields in a non-minimal way, a 4 in general depends on the background Riemann tensor and gauge field strengths.A surprising fact, observed in [39] and later verified in [40,41], is that for N ≥ 2 supergravity, for which we can divide the massless fields as belonging to n H hypermultiplets, n V vector multiplets, one gravity multiplet and (N − 2) gravitino multiplet, a 4 is proportional to the Euler density: In particular, the dependence on the background gauge fields can be expressed in terms of the Ricci tensor after using Einstein's equation and the equations of motion and the Bianchi identities of the gauge field strengths.Since the Euler density is a topological term, its integral over the near horizon AdS 2 × S 2 geometry and the full geometry gives the same result (after adding appropriate boundary terms for AdS 2 ×S 2 ).This has been explicitly verified in appendix A. In both cases d 4 x √ g E 4 = 64π 2 and (4.10) gives Since (4.12) is the same for the zero temperature and finite temperature geometry, we only need to check if the zero mode contributions agree between the two computations.

Zero mode contribution
Zero modes are generated by symmetry transformations of the asymptotic geometry that are broken by the solution.The transformations parameters are non-normalizable but the deformations they produce are normalizable.We shall compare the zero mode contributions from each super-multiplet separately, since our goal is to establish that the agreement between the calculations at zero temperature and finite temperature agree for an arbitrary N ≥ 2 supergravity.

Hypermultiplet zero modes
The hypermultiplet fields, consisting of scalars and spin half fermions, do not have any zero modes either in the zero temperature AdS 2 × S 2 geometry or in the finite temperature geometry.Therefore their contribution to the logarithmic corrections agree trivially between zero temperature computation and finite temperature computation.

Vector multiplet zero modes
The gauge fields of the vector multiplet have zero modes in the zero temperature geometry, but the contribution from the zero modes turns out to be the same that would have been there if these modes had been non-zero modes [25].Therefore the heat kernel already captures their contribution and no further corrections are necessary (see footnote 4).For the finite temperature geometry there are no zero modes for the vector multiplet fields [13].Therefore the contribution from the vector multiplet zero modes also agree trivially between the zero and finite temperature calculations.

Gravity multiplet zero modes
Next we turn to the gravity multiplet zero modes.In the zero temperature geometry there are both fermionic and bosonic zero modes.In particular we know from eq.( 5.31) of [25] that the net logarithmic contribution to the entropy from the gravitino zero modes is given by 4 ln A and from eqs.(2.19) and (4.39) of [25] that the net logarithmic correction due to the metric and graviphoton zero modes is given by −3 ln A. These factors include the terms that need to be subtracted from the heat kernel contribution due to the fact that the heat kernel includes contributions from the zero modes treating them as non-zero modes.Therefore the net logarithmic corrections due to the zero modes in the zero temperature geometry is ln A .
We shall now carry out the analysis in the finite temperature geometry.Eq.(2.41) of [13] tells us that in D = 4 there is no logarithmic correction from the zero modes of the metric.
There is, however, one subtle point we need to take into account.A priori a rotating black hole in the Lorentzian geometry carries three translational zero modes generated by translation along the three spatial directions and two rotational zero modes corresponding to rotations about the two axes perpendicular to the axis of rotation.However in the Euclidean geometry, corresponding to a black hole rotating around the third axis with angular velocity Ω 3 at the horizon, we require the fields to be periodic under simultaneous translation of the Euclidean time τ by β and the azimuthal angle ϕ by −iβ Ω 3 .For generic β Ω 3 , it was argued in [13] that only the translational zero mode along the z direction satisfies this periodicity requirement.
The other two translational modes and the rotational modes have non-trivial dependence on ϕ and fails to satisfy the periodicity requirement.However, in the present context we have β Ω 3 = −2πi and so the modes are required to be periodic under τ → τ + β, ϕ → ϕ − 2π.
Since the modes are τ independent, this only demands that they are periodic under 2π shift of ϕ and this is automatically satisfied.This has been illustrated in section 3 for the rotational zero modes.This means that we now need to take into account the contribution from two extra translation zero modes and two extra rotational zero modes.
Now it was shown in [13] that in D space-time dimensions the contributions from the translational zero modes are proportional to (D − 4).Therefore they do not contribute for D = 4. 5 This leaves us with the two rotational zero modes described in section 3.In order to evaluate their contribution we use (2.19) to conclude that if we expand the metric fluctuation µν are basis functions, normalized so that they do not carry any a dependence, then the integration measure over the modes h n should come as n dh n . (4.14) In particular there is no a dependence of the measure.For the zero modes, we can now change variable of integration from h n to the parameters α x , α y describing rotational zero modes.As described in the last paragraph of section 3, for each zero mode, this introduces a jacobian factor of a 2 .Since the α x , α y integration ranges are a independent, it produces a net factor of (a 2 ) 2 .On the other hand, the heat kernel result counts the contribution from each mode (including the zero modes) as a, since the non-vanishing eigenvalues of the kinetic operator are of order 1/a 2 .Therefore we need to divide the partition function by a factor of a for each zero mode h n .This yields a net contribution of ln (a 2 ) 2 a 2 = 2 ln a ∼ ln A , ( to the entropy. We now turn to the contribution due to the gravitino zero modes of the gravity multiplet.As described in eq.(2.21), if we expand the gravitino field ψ µ as n ψ n g Naively integral over these gravitino zero modes will vanish but this is avoided as follows.As mentioned in the introduction, for every pair of broken supersymmetry, we need to insert a factor of 2J 3 in the path integral.Since a supersymmetric black hole has four unbroken supersymmetries and since N = 2 supergravity has eight unbroken supersymmetries in Minkowski space, the black hole breaks four of these eight supersymmetries.So we have a net factor of (2J 3 ) 2 .We shall now show that the presence of this factor of (2J 3 ) 2 makes the zero mode integral non-vanishing.
We shall construct J 3 using the Noether procedure.First note that the action has the form where E ρ a is the inverse vierbein.Since J 3 is the generator of rotation, it acts on the field ψ without any factor of a. Therefore the contribution to J 3 from the fermions, obtained using the Noether procedure from the action (4.17), is a linear combination of terms of the form a ψ m ψ n .If we denote by ψ 1 , • • • , ψ 4 the four fermion zero modes then that the relevant part of 16) we now see that the integration over the zero modes give (4.18) We need to divide this by the power of a that is generated by the contribution of the zero modes to the heat kernel since the heat kernel treats the zero modes as if they were non-zero modes.Indeed, the expression based on the heat kernel counts a factor of a −1 for each pair of gravitino zero modes, since this is how the non-zero eigenvalues of the Dirac operator scales.
Therefore the heat kernel contribution includes a net factor of a −2 from the four gravitino zero modes which need to be removed from the expression for heat kernel contribution to the index.Dividing (4.18) by a −2 , we see that the net excess contribution due to the gravitino zero modes, that is not counted in the heat kernel analysis, is independent of a. Therefore it gives zero net logarithmic correction to the entropy.
Combining this result with (4.15) we get the net logarithmic contribution to the index from the gravity multiplet zero modes to be ln A .This agrees with the result (4.13) of the zero temperature analysis.

Gravitino multiplet zero modes
Finally we turn to the extra gravitino multiplets that are present in N > 2 supergravity theories.These have no zero modes in the zero temperature sector, as can be seen from the analysis of the zero mode structure for black holes in N = 4 and N = 8 supergravity.In the finite temperature geometry they lead to 4(N − 2) fermion zero modes that have the following origin.Since a supersymmetric black hole preserves four supersymmetries and an N ≥ 2 supergravity has 4N unbroken supersymmetries in Minkowski space, 4(N −1) supersymmetries are broken by the black hole background.This leads to 4(N − 1) goldstino fermion zero modes.Of these four fermion zero modes have already been attributed to the zero modes of the gravitinos belonging to the gravity multiplet of N = 2 supergravity.The remaining 4(N − 2) zero modes must come from the (N − 2) extra gravitino multiplets.The analysis of logarithmic contribution due to these zero modes follows an analysis identical to that for the gravitino zero modes coming from the gravity multiplet, and we are led to the same conclusion that they do not give any additional logarithmic contribution besides those already counted in the heat kernel.
This shows that the contribution to the logarithmic correction in the zero temperature and finite temperature backgrounds agree for each super-multiplet separately.

Conclusion
In this paper we have used the formalism developed in [19] to compute logarithmic corrections to the logarithm of supersymmetric black hole index by working with a finite temperature solution.The result is in perfect agreement with the zero temperature, near horizon computation of the same quantity.
Since the two procedures are different, the equality of the final results gives confirmation of both methods.This also gives additional confirmation of the final results for the logarithmic corrections, many of which have been tested in string theory via microscopic counting.Furthermore, since the finite temperature method for computing the index is closely related to the computation of the logarithmic correction to the entropy of non-supersymmetric black holes, the result of this paper also gives confidence in the results for non-supersymmetric black holes, for which there is as yet no independent verification of the results from microscopic counting of states.A Comparing the heat kernel contributions in the zero and finite temperature computations In this appendix, we describe explicit checks on the equality between the heat kernel contributions in the zero temperature computation and the finite temperature computation.
In the usual literature, the metric for the Kerr-Newman solution is expressed in terms of Boyer-Lindquist coordinates described in (2.1).For our purpose, we find that it is easy to compute the curvature invariants like R µν R µν , R µνρσ R µνρσ , F µν F µν , etc. by going to the null-Kerr coordinates which are defined by the following coordinate transformations The inverse metric is This form of g µν makes the computation of the curvature invariants easy.It is now simple to check that we recover the result of R µν R µν and R µνρσ R µνρσ for Kerr-Newman geometry as given in [42][43][44], where r 0 is the upper limit of the r-integral which we eventually take to infinity.The term proportional to r 0 is removed by adding an appropriate counterterm proportional to the length of the boundary since this does not contribute to the ground state degeneracy [22].The finite part of the result, that contributes to the log of the degeneracy, is the same as in the Kerr-Newmann geometry given in (A.8).
functions, normalized so that they do not carry any a dependence, then the measure over the gravitino modes should come as n d(aψ n ) .(4.16)

Acknowledgement:
We thank Alok Laddha for useful discussions.Research at ICTS-TIFR is supported by the Department of Atomic Energy Government of India, under Project Identification No. RTI4001.The work of A.S. is supported by ICTS-Infosys Madhava Chair Professorship and the J. C. Bose fellowship of the Department of Science and Technology.